On the 2-D Riemann problem for the compressible Euler equations I. Interaction of shocks and rarefaction waves

Abstract

We are concerned with the Riemann problem for the two-dimensional com-pressible Euler equations in gas dynamics. The central point at this issue is the dynamical interaction of shock waves, centered rarefaction waves, and contact discontinuities that connect two neighboring constant initial states in the quadrants. The Riemann problem is classified into eighteen genuinely different cases. For each configuration, the structure of the Riemann solution is analyzed using the method of characteristics, and corresponding numerical solution is illustrated by contour plots using an upwind averaging scheme that is second order in the smooth region of the solution. In the first paper we mainly focus on the interaction of shocks and rarefaction waves. The theory is developed from an analysis of the structure of the Euler equations and their Riemann solutions in [CC, ZZ] and the MmB scheme [WY].